To find the value of $x$, we'll use the fact that the angles around a point add up to $360^\circ$ and that vertically opposite angles are equal.

Let's start by analyzing the angles:

1. The angle $(3y + 10)^\circ$ is vertically opposite the angle $(9x + 1)^\circ$. Therefore, we have: $3y + 10 = 9x + 1$ Simplifying this equation: $3y - 9x = -9 \quad \text{(Equation 1)}$

2. The angle $(5y - 24)^\circ$ is vertically opposite the angle $82^\circ$. Therefore: $5y - 24 = 82$ Simplifying this equation: $5y = 106$ $y = 21.2 \quad \text{(Equation 2)}$

Now that we have $y = 21.2$, we can substitute this value into Equation 1 to find $x$.

$3(21.2) - 9x = -9$ $63.6 - 9x = -9$ $-9x = -9 - 63.6$ $-9x = -72.6$ $x = \frac{72.6}{9} = 8.0667$

Thus, the value of $x$ is approximately $8.067$.

Would you like further details or have any other questions? Here are some related questions you might be interested in:

1. How can vertically opposite angles be used to solve more complex problems?
2. What are the steps to solve simultaneous equations in general?
3. How does the sum of angles around a point help in geometry problems?
4. Can you explain how to check if two angles are vertically opposite?
5. What are some common mistakes when solving problems with angles?

Tip: Always double-check the equality of vertically opposite angles before solving, as this can simplify the problem significantly.